Obtaining interpretable parameters from reparameterizing longitudinal models: transformation matrices between growth factors in two parameter spaces

The linear spline growth model (LSGM), which approximates complex patterns using at least two linear segments, is a popular tool for examining nonlinear change patterns. Among such models, the linear-linear piecewise change pattern is the most straightforward one. An earlier study has proved that other than the intercept and slopes, the knot (or change-point), at which two linear segments join together, can be estimated as a growth factor in a reparameterized longitudinal model in the latent growth curve modeling framework. However, the reparameterized coefficients were no longer directly related to the underlying developmental process and therefore lacked meaningful, substantive interpretation, although they were simple functions of the original parameters. This study proposes transformation matrices between parameters in the original and reparameterized models so that the interpretable coefficients directly related to the underlying change pattern can be derived from reparameterized ones. Additionally, the study extends the existing linear-linear piecewise model to allow for individual measurement occasions, and investigates predictors for the individual-differences in change patterns. We present the proposed methods with simulation studies and a real-world data analysis. Our simulation studies demonstrate that the proposed method can generally provide an unbiased and consistent estimation of model parameters of interest and confidence intervals with satisfactory coverage probabilities. An empirical example using longitudinal mathematics achievement scores shows that the model can estimate the growth factor coefficients and path coefficients directly related to the underlying developmental process, thereby providing meaningful interpretation. For easier implementation, we also provide the corresponding code for the proposed models.